# The ac method. How to factorise a quadratic expression using the ac factoring method.

The ac method can be used for factorising expressions of the form ax² + bx + c. To factorise a quadratic using the ac method follow these 5 steps:

1) Multiply a and c together.

2) Now look for the factor pairs of ac that sum to give b.

3) Next rewrite your quadratic splitting up the middle term using the numbers that you have just found.

4) Factorise the first two terms of the quadratic into a single bracket and the last two terms into a second bracket.

5) Finally group these together to give the final answer.

Example 1

Factorise 3x² + x – 10

1) Multiply a and c together

a= 3 (number before x²) and c = -10 (number at the end) so 3 × - 10 = -30

2) Now look for the factor pairs of ac that sum to give b.

The factor pairs of -30 are:

-1 × 30

+ 1 × -30

-2 × 15

2 × -15

3 × -10

-3 × 10

5 × -6

-5 × 6

Now since b = 1 (number before x) then the factor pair needed is -5 and 6 (since -5 + 6 = 1)

3) Next rewrite your quadratic splitting up the middle term using the numbers that you have just found.

3x² + x – 10 = 3x² -5x + 6x – 10

4) Factorise the first two terms of the quadratic into a single bracket and the last two terms into a second bracket.

3x² -5x + 6x – 10

x(3x-5) + 2(3x-5)

5) Finally group these together to give the final answer.

(x+2)(3x-5)

Example 2

Factorise 8x² -14x + 6

1) Multiply a and c together

a= 8 (number before x²) and c = 6 (number at the end) so 8 × 6 = 48

2) Now look for the factor pairs of ac that sum to give b.

The factor pairs of -30 are:

1 × 48

2 × 24

3 × 16

4 × 12

6 × 8

-1 × -48

-2 × -24

-3 × -16

-4 × -12

-6 × -8

Now since b = -14 (number before x) then the factor pair needed is -6 and -8 (since -6 + -8 = -14)

3) Next rewrite your quadratic splitting up the middle term using the numbers that you have just found.

8x² -14x + 6 = 8x² -6x - 8x + 6

4) Factorise the first two terms of the quadratic into a single bracket and the last two terms into a second bracket.

8x² -6x - 8x + 6

2x(4x-3) - 2(4x-3)

5) Finally group these together to give the final answer.

(2x-2)(4x-3)

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